Theorems · Theorem · harmonic analysis
UnitAddTorus.hasSum_mFourier_series_L2
∀ {d : Type u_1} [inst : Fintype d] (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
HasSum (fun i => UnitAddTorus.mFourierCoeff (↑↑f) i • UnitAddTorus.mFourierLp 2 i) fThe Fourier series of an L2 function f sums to f in the L² norm.
- Defined in
- Mathlib.Analysis.Fourier.AddCircleMulti
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 274 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Fintype
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- ENNRealstatement · cited by 9,879
- Fintypestatement and proof · cited by 7,736
- Complexstatement and proof · cited by 5,565
- AddSubgroupstatement · cited by 3,232
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- MeasureTheory.AEEqFunstatement and proof · cited by 856
- MeasureTheory.Lpstatement and proof · cited by 715
- HasSumstatement and proof · cited by 518
- MeasureTheory.AEEqFun.caststatement and proof · cited by 380
- UnitAddCirclestatement · cited by 157
Cited by1
Results whose statement or proof uses this declaration.
- UnitAddTorus.hasSum_mFourier_series_of_summableproof · cited by 1